The spontaneous breaking of the Becchi-Rouet-Stora-Tyutin (BRST) fermionic symmetry treated as generalized supersymmetry is studied on the basis of a simple non-gauge model invariant under BRST supersymmetry. The Nambu-Goldstone theorem is stated and the spontaneous breakdown of the BRST supersymmetry is illustrated by the above non-gauge model. This indicates that the dynamical stability of BRST supersymmetry in non-abelian gauge theories should also be carefully examined, if one takes the Faddeev-Popov Lagrangian as a basis of the formal canonical treatment.